Advertisements
Advertisements
Question
Write the value \[\left( \hat{ i } \times \hat{ j } \right) \cdot \hat{ k } + \hat{ i } \cdot \hat{ j } .\]
Advertisements
Solution
\[\left( \hat{ i } \times \hat{ j } \right) . \hat{ k } + \hat{ i } . \hat{ j } \]
\[ = \hat{ k } .\hat{ k } + 0\]
\[ = \left| \hat{ k } \right|^2 + 0\]
\[ = 1^2 + 0 ( \because \left| k \right| = 1)\]
\[ = 1\]
APPEARS IN
RELATED QUESTIONS
Find `|veca × vecb|`, if `veca = hati - 7hatj + 7hatk` and `vecb = 3hati - 2hatj + 2hatk`.
Find a unit vector perpendicular to each of the vector `veca + vecb` and `veca - vecb`, where `veca = 3hati + 2hatj + 2hatk` and `vecb = hati + 2hatj - 2hatk`.
Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).
Area of a rectangle having vertices A, B, C, and D with position vectors `-hati + 1/2 hatj + 4hatk, hati + 1/2 hatj + 4hatk, and -hati - 1/2j + 4hatk,` respectively is ______.
\[\text{ If } \vec{a} = \hat { i } + 3 \hat { j } - 2 \hat { k } \text{ and } \vec{b} = - \hat { i } + 3 \hat { k } , \text{ find } \left| \vec{a} \times \vec{b} \right| .\]
If \[\vec{a} = 3 \hat { i } + 4 \hat { j } \text{ and } \vec{b} = \hat { i } + \hat{ j } + \hat{ k } ,\] find the value of \[\left| \vec{a} \times \vec{b} \right| .\]
Find a unit vector perpendicular to the plane ABC, where the coordinates of A, B and Care A (3, −1, 2), B (1, −1, −3) and C (4, −3, 1).
If a, b, c are the lengths of sides, BC, CA and AB of a triangle ABC, prove that \[\vec{BC} + \vec{CA} + \vec{AB} = \vec{0}\] and deduce that \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} .\]
Let \[\vec{a} = \hat{ i } + 4 \hat{ j } + 2 \hat{ k } , \vec{b} = 3 \hat{ i }- 2 \hat{ j } + 7 \hat{ k } \text{ and } \vec{c} = 2 \hat{ i } - \hat{ j } + 4 \hat{ k } .\] Find a vector \[\vec{d}\] which is perpendicular to both \[\vec{a} \text{ and } \vec{d}\] \[\text{ and } \vec{c} \cdot \vec{d} = 15 .\]
If \[\vec{a} = 2 \hat{ i } - 3 \hat{ j } + \hat{ k } , \vec{b} = -\hat{ i } + \hat{ k } , \vec{c} = 2 \hat{ j } - \hat{ k } \] are three vectors, find the area of the parallelogram having diagonals \[\left( \vec{a} + \vec{b} \right)\] and \[\left( \vec{b} + \vec{c} \right)\] .
Using vectors, find the area of the triangle with vertice A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) .
Using vectors, find the area of the triangle with vertice A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1) .
Write the value of \[\hat{ i } . \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } . \left( \hat{ k } \times \hat{ i } \right) + \hat{ k } . \left( \hat{ i } \times \hat{ j } \right) .\]
If \[\vec{a} \text{ and } \vec{b}\] are two vectors of magnitudes 3 and \[\frac{\sqrt{2}}{3}\] espectively such that \[\vec{a} \times \vec{b}\] is a unit vector. Write the angle between \[\vec{a} \text{ and } \vec{b} .\]
Write the value of \[\hat{ i } \times \left(\hat{ j } \times \hat{ k } \right) .\]
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors such that \[\vec{a} \times \vec{b}\] is also a unit vector, find the angle between \[\vec{a} \text{ and } \vec{b}\] .
If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} . \vec{b} \right| = \left| \vec{a} \times \vec{b} \right|,\] write the angle between \[\vec{a} \text{ and } \vec{b} .\]
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then write the value of \[\left| \vec{a} \times \vec{b} \right|^2 + \left( \vec{a} . \vec{b} \right)^2 .\]
Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] with magnitudes 1 and 2 respectively and when \[\left| \vec{a} \times \vec{b} \right| = \sqrt{3} .\]
Find λ, if \[\left( 2 \hat{ i } + 6 \hat{ j } + 14 \hat{ k } \right) \times \left( \hat{ i } - \lambda \hat{ j } + 7 \hat{ k } \right) = \vec{0} .\]
Write the value of \[\left( \hat{ i } \times \hat{ j } \right) \cdot \hat{ k } + \left( \hat{ j } + \hat{ k } \right) \cdot \hat{ j } \]
If \[\vec{a}\] is any vector, then \[\left( \vec{a} \times \hat{ i } \right)^2 + \left( \vec{a} \times \hat{ j } \right)^2 + \left( \vec{a} \times \hat{ k } \right)^2 =\]
The vector \[\vec{b} = 3 \hat { i }+ 4 \hat {k }\] is to be written as the sum of a vector \[\vec{\alpha}\] parallel to \[\vec{a} = \hat {i} + \hat {j}\] and a vector \[\vec{\beta}\] perpendicular to \[\vec{a}\]. Then \[\vec{\alpha} =\]
If \[\vec{a} = \hat{ i } + \hat{ j } - \hat{ k } , \vec{b} = - \hat{ i } + 2\hat{ j } + 2 \hat{ k } \text{ and } \vec{c} = - \hat{ i } + 2 \hat{ j } - \hat{ k } ,\] then a unit vector normal to the vectors \[\vec{a} + \vec{b} \text{ and } \vec{b} - \vec{c}\] is
A unit vector perpendicular to both \[\hat{ i } + \hat{ j } \text{ and } \hat{ j } + \hat{ k } \] is
The value of \[\left( \vec{a} \times \vec{b} \right)^2\] is
If θ is the angle between any two vectors `bara` and `barb` and `|bara · barb| = |bara xx barb|` then θ is equal to ______.
The number of vectors of unit length perpendicular to the vectors `vec"a" = 2hat"i" + hat"j" + 2hat"k"` and `vec"b" = hat"j" + hat"k"` is ______.
What is the sum of vector `veca = hati - 2hati + hatk, vecb = - 2hati + 4hatj + 5hatk` and `vecc = hati - 6hatj - 7hatk`
Find the area of the parallelogram whose diagonals are `hati - 3hatj + hatk` and `hati + hatj + hatk`.
If `veca` and `vecb` are two non-zero vectors such that `|veca xx vecb| = veca.vecb`, find the angle between `veca` and `vecb`.
